You're thinking about maths, logic and algorithmic, abstract disciplines that go nicely with programming, coding, or however you would like to call the physical implementation of the mathematical ideas derived using your pencil. Although you could argue about when exactly the step is made from abstraction to implementation, I like to think that it's somewhere between compile time and throughout testing.
They were programs in every sense of the word. Back then it was often QuickBasic or x86 assembly language, nowadays they tend to be either Lua or some Lisp (usually Scheme). I actually find the process helpful, because when it's not as convenient to just run the program to see if it works, I find that I reason about the programs more deeply. YMMV, of course.
